用变量分离常微分方程方法求Boussinesq-double sine-Gordon 和Boussinesq-double sinh-Gordon方程的行波解

The variable separated ODE method for travelling wave solutions for the Boussinesq-double sine-Gordon and the Boussinesq-double sinh-Gordon equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA Received 7 May 2005; accepted 8 March 2006 Available online 6 May 2006 Abstract Travelling wave solutions for the Boussinesq-double sine-Gordon (B-sine-Gordon) equation, the Boussinesq-double sinh-Gordon equation (B-sinh-Gordon), and the Boussinesq-Liouville (BL) equation are formally derived. The approach rests mainly on the variable separated ODE method. Distinct sets of exact solitary wave solutions, that possess distinct physical structures, are obtainedfor each equation. © 2006 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction 1. Introduction It is well-known that the linear Boussinesq equation is a fourth order PDE given by utt − αuxx + uxxxx = 0, (1) that includes the physical dispersion term. The sine-Gordon equation is a second order nonlinear PDE given by utt − uxx + sin u = 0. (2) This equation appeared first in the study of differential geometry of surfaces [1,9,2,7,16,3,8,15]withGaussian curvature K =−1. The sine-Gordon equation is a completely integrable, infinite dimensional Hamiltonian system [1], and it can be solved by the inverse scattering method. Integrable means that there is a sufficiently large number of conserved quantities [1]. The term sin u is the Josephson current across an insulator between two superconductors [7]. The double sine-Gordon equation utt − uxx + sin u + sin 2u = 0, (3) 2 A.-M. Wazwaz / Mathematics and Computers in Simulation 72 (2006) 1–9 appears in many scientific applications. However, the sinh-Gordon equation utt − uxx + sinh u = 0, (4) appears in integrable quantum field theory, kink dynamics, and fluid dynamics [1,9,2,7,16,3,8,15]. The sinh-Gordon equation is completely integrable because it possesses similarity reductions to third Painlev´ e equation. Moreover, the double sinh-Gordon equation utt − uxx + sinh u + sinh 2u = 0, (5) has lot of scientific applications as well. It is well-known that searching for explicit solutions for nonlinear evolution equation, by using different methods, is the goal for many researchers. Many powerful methods, such as B¨ acklund transformation, inverse scattering method, Hirota bilinear forms, pseudo spectral method, the tanh-sech method [4–6,12,11], the sine–cosine method [10], and many other techniques were successfully used to investigate these types of equations. Practically, there is no unified method that can be used to handle all types of nonlinear problems. In this work, the Boussinesq-double sine-Gordon (B-sine-Gordon) equation, the Boussinesq-double sinh-Gordon equation (Boussinesq-sinh-Gordon), and the Boussinesq-Liouville Eqs. (1) and (2) (BL-I) and (BL-II) given by utt − αuxx + uxxxx = sin u + 3 2 sin 2u, (6) utt − αuxx + uxxxx = sinh u + 3 2 sinh 2u, (7) utt − αuxx + uxxxx = eu + 3 4 sinh e2u, (8) and utt − αuxx + uxxxx = e−u + 3 4 e−2u, (9) respectively, will be investigated. The aforementioned equations were established by combining the linear Boussinesq equation with the double sine-Gordon, double sinh-Gordon, and the Liouville equations. The objectives of this work are two folds. The first goal is to conduct an analysis on these equations to derive more exact solitary wave solutions. Secondly, we aim to emphasize the power of the variable separated ODE method that will be employed here. This method is developed by Sirendaoreji et al. in [9], used by Fu et al. in [2] and by Wazwaz [13,14]. It works effectively if the equation involves sine, cosine, hyperbolic sine, and hyperbolic cosine functions. In what follows we highlight the main features of the method as introduced in [9] where more details and examples can